(New page: == Linearity == A system is called linear if it's output can be shown as a sum of it's inputs prior to being input to the system. == Example of a Linear System == == Example of a Non-Li...) |
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== Example of a Linear System == | == Example of a Linear System == | ||
| + | Given the system <math>y(t) = 5x(t)</math> | ||
| + | If we apply two input signals <math>x_1(t) = 4n</math> and <math>x_2(t) = 5</math>, we end up with two outputs <math>y_1(t) = 20n</math> and <math>y_2(t) = 25</math>. | ||
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| + | If we apply the sum of the two inputs <math>x_3(t) = 4n + 5</math>, we end up with the output <math>y_3(t) = 20n + 25</math>. | ||
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| + | Since <math>y_1(t) + y_2(t) = y_3(t)</math> the system is linear. | ||
== Example of a Non-Linear System == | == Example of a Non-Linear System == | ||
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If we apply the sum of the two inputs <math>x_3[n] = 4x + 5</math>, we end up with the output <math>y_3[n] = 16x^2 + 40x + 25</math>. | If we apply the sum of the two inputs <math>x_3[n] = 4x + 5</math>, we end up with the output <math>y_3[n] = 16x^2 + 40x + 25</math>. | ||
| − | Since <math>y_1 + y_2 \ne y_3</math> the system is <b>NOT</b> linear. | + | Since <math>y_1[n] + y_2[n] \ne y_3[n]</math> the system is <b>NOT</b> linear. |
Latest revision as of 16:38, 11 September 2008
Linearity
A system is called linear if it's output can be shown as a sum of it's inputs prior to being input to the system.
Example of a Linear System
Given the system $ y(t) = 5x(t) $
If we apply two input signals $ x_1(t) = 4n $ and $ x_2(t) = 5 $, we end up with two outputs $ y_1(t) = 20n $ and $ y_2(t) = 25 $.
If we apply the sum of the two inputs $ x_3(t) = 4n + 5 $, we end up with the output $ y_3(t) = 20n + 25 $.
Since $ y_1(t) + y_2(t) = y_3(t) $ the system is linear.
Example of a Non-Linear System
Given the system $ y[n] = x[n]^{2} $
If we apply two input signals $ x_1[n] = 4x $ and $ x_2[n] = 5 $, we end up with two outputs $ y_1[n] = 4x^2 $ and $ y_2[n] = 25 $.
If we apply the sum of the two inputs $ x_3[n] = 4x + 5 $, we end up with the output $ y_3[n] = 16x^2 + 40x + 25 $.
Since $ y_1[n] + y_2[n] \ne y_3[n] $ the system is NOT linear.
